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Kronecker :: rationalNormalForm

rationalNormalForm -- rational normal form of a matrix

Synopsis

Description

This function produces a matrix B in rational normal form, and invertible matrices P and Q such that P*Q = I and B = P*A*Q.
i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing
i2 : M = R^4

      4
o2 = R

o2 : R-module, free
i3 : A = random(M,M)

o3 = | -44 -46 43  39  |
     | -21 14  -28 23  |
     | 0   22  48  -5  |
     | -18 -20 -34 -17 |

             4       4
o3 : Matrix R  <--- R
i4 : factor det(x*id_M - A)

                       2
o4 = (x + 22)(x + 34)(x  + 44x + 17)

o4 : Expression of class Product
i5 : (B,P,Q) = rationalNormalForm A

o5 = (| 1 0 0   0 |, | 30  -12 -39 31  |, | -37 30  50  19 |)
      | 0 1 0   0 |  | 3   38  29  -48 |  | -14 28  41  19 |
      | 0 0 -44 1 |  | -22 50  -27 12  |  | -42 -9  -39 1  |
      | 0 0 -17 0 |  | 47  48  14  44  |  | -3  -23 -49 0  |

o5 : Sequence
i6 : B - P*A*Q == 0

o6 = true
i7 : P*Q - id_M == 0

o7 = false

Ways to use rationalNormalForm :